We say that f is linear provided that for every x, y in its domain,
f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the
set of real numbers), then f is defined by f(x) = cx for some c belong to
R.

How do you determine the radius of the circle that maximizes the area of an irregular
n-gon circumscribed on it? With the Pari computer algebra system, Doctor Vogler
approaches the question using numerical techniques such as Newton's Method and a
binary search, which suggests that no closed-form expression exists.

A student wants to write a computer program to find the minimum number
of stamps needed to create a desired postage given the various stamp
values that are currently available. Doctor Tom explains why that sort
of problem presents a very difficult challenge.

What kind of information could you give all 10 people such that if any
3 of them were to get together, they would be able to launch the
missiles, but if there were only 2 of them, the information would be
insufficient to figure out the code?

The proof of Fermat's Last Theorem shows how the L-series of elliptic
curves and the M-series of modular forms correspond, and how modular
forms are composed of different "ingredients." Can you provide a
description of the ingredients of a modular form?

A student with six linear equations in two variables wonders if the Euclidean
algorithm would solve it. Doctor Vogler simplifies the system by applying the Extended
Euclidean Algorithm, and introducing the Chinese Remainder Theorem.